The Loop Algorithm

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  • 21 July 1997
Abstract
A review of the Loop Algorithm, its generalizations and its applications is given, including some new results. The loop algorithm is a Monte Carlo procedure which performs nonlocal changes of worldline configurations, determined by local stochastic decisions. It is based on a formulation of physical models in an extended ensemble of worldlines and graphs, and is related to Swendsen-Wang cluster algorithms. It overcomes many of the difficulties of traditional worldline simulations. Autocorrelations between successive Monte Carlo configurations are reduced by orders of magnitude. The grand-canonical ensemble (e.g. varying winding numbers) is naturally simulated. The continuous time limit can be taken directly. Improved Estimators exist which further reduce the errors of measured quantities. The algorithm remains unchanged in any dimension and for varying bond-strengths. It becomes less efficient in the presence of strong site disorder or strong magnetic fields. It applies directly to locally XYZ-like spin, fermion, and hard-core boson models. It has been extended to the Hubbard and the tJ model and generalized to higher spin representations. There have already been several large scale applications, especially for Heisenberg-like models, including a high statistics continuous time calculation of quantum critical exponents on a regularly depleted two-dimensional lattice of up to 20000 spatial sites at temperatures down to T=0.01 J.

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